Preheat the oven to 375°F.
Line a 15-by-10-inch jelly roll pan with parchment or grease it.
Sift the cake flour and baking powder together twice and set aside.
In a large bowl, cream the butter and powdered sugar together until very light and fluffy.
Add the egg yolks one at a time until all are incorporated, scraping the bowl periodically with a rubber spatula.
Fold the melted chocolate and vanilla extract into the egg mixture until fully incorporated.
In a large bowl, whip the egg whites until frothy.
Gradually beat in the granulated sugar to form stiff, glossy peaks.
Stir one third of the meringue into the chocolate mixture to lighten it, then fold in the remaining meringue.
Fold the flour mixture into the chocolate mixture in fourths, until just blended.
Spread the batter as evenly as possible in the prepared pan.
Bake the cake for about 12 minutes, or until a toothpick inserted near the center of the cake comes out clean.
Let cool in the pan, then remove to a cutting board.
Cut the cake into three 5-by-10-inch strips.
Brush all sides of each strip with the rum syrup.
Spread the chocolate cream on top of the bottom and middle layers and stack into a 3-layer cake.
Refrigerate for at least 2 hours before serving.